1. Discrete Mathematics University of Jazeera College of Information Technology & Design Khulood Ghazal Connectivity Lecture _13

2. Paths in Undirected Graphs There is a path from vertex v 0 to vertex v n if there is a sequence of edges from v 0 to v n –This path is labeled as v 0,v 1,v 2,…,v n and has a length of n. The path is a circuit if the path begins and ends with the same vertex. A path is simple if it does not contain the same edge more than once.

3. Paths in Undirected Graphs A path or circuit is said to pass through the vertices v 0, v 1, v 2, …, v n or traverse the edges e 1, e 2, …, e n.

4. Example u 1, u 4, u 2, u 3 –Is it simple? –yes –What is the length? –3 –Does it have any circuits? –no u 1 u 2 u 5 u 4 u 3

5. Example u 1, u 5, u 4, u 1, u 2, u 3 –Is it simple? –yes –What is the length? –5 –Does it have any circuits? –Yes; u 1, u 5, u 4, u 1 u 1 u 2 u 5 u 3 u 4

6. Example u 1, u 2, u 5, u 4, u 3 –Is it simple? –yes –What is the length? –4 –Does it have any circuits? –no u 1 u 2 u 5 u 3 u 4

7. Connectedness An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph. There is a simple path between every pair of distinct vertices of a connected undirected graph.

8. Example Are the following graphs connected? c e f a d b g e c a f b d YesNo

9. Cut edges and vertices If one can remove a vertex (and all incident edges) and produce a graph with more connected components, the vertex is called a cut vertex. If removal of an edge creates more connected components the edge is called a cut edge or bridge.

10. Example Find the cut vertices and cut edges in the following graph. a b c d f e h g Cut vertices: c and e Cut edge: (c, e)

11. Connectedness in Directed Graphs A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph. A directed graph is weakly connected if there is a path between every pair of vertices in the underlying undirected graph. A directed graph is weakly connected iff there is always a path between two vertices when the direction are disregarded.

12. Example Is the following graph strongly connected? Is it weakly connected? a b c e d This graph is strongly connected. Why? Because there is a directed path between every pair of vertices. If a directed graph is strongly connected, then it must also be weakly connected.

13. Example Is the following graph strongly connected? Is it weakly connected? a b c e d This graph is not strongly connected. Why not? Because there is no directed path between a and b, a and e, etc. However, it is weakly connected.